An Alexandre Lacazette goal in the final minute of the match ensured that the game went the distance, and it was the Gunners who held their nerve in a tight shootout at the Aviva Stadium.
German centre-back Antonio Rudiger had powered home a header with less than six minutes on the clock for what looked like being the game's only goal, but the Blues were pegged back at the death and suffered defeat under Maurizio Sarri.
Gunners manager Unai Emery, himself embarking on a new chapter at the London club, watched on as his side edged the eighth meeting in 13 months between these two sides, taking Arsenal to six points from their three matches in the prestige friendly competition.
Ramsey's place in the starting lineup was taken by Emile Smith Rowe, and it was the youngster - not helped by the late change - who appeared to lose Rudiger for the game's opening goal, with the centre-back directing Cesc Fabregas's header out of Petr Cech's reach.
Chelsea's fast start should have been rewarded with a second goal, only for Alvaro Morata to meet his match in Cech with a weak penalty after Callum Hudson-Odoi forced Hector Bellerin into making a challenge inside the box.
Hudson-Odoi maintained his impressive pre-season form throughout the first half, slotting Morata through on goal less than 10 minutes after the penalty miss, with the Spaniard once again being denied by Cech down low.
That Morata miss proved to be the catalyst for Arsenal to finally spark into life, and they really should have been on level terms 35 minutes in when a swift move culminated in Henrikh Mkhitaryan spotting Pierre-Emerick Aubameyang at the back post.
Aubameyang failed to make any contact with the ball from a glorious position, though, having been put off by Chelsea skipper Cesar Azpilicueta.
Sead Kolasinac then squandered a good chance, moments before Hudson-Odoi burst through up the other end and had his shot kept out by a mixture of Cech and Shkodran Mustafi in what was proving to be an end-to-end affair.
Hudson-Odoi's audition for a starting spot in this weekend's Community Shield continued into the second half, as the youngster forced another save out of Cech and then played a part in winning a corner that Rudiger this time headed wide.
In stark contrast to the first half, the second period was bereft of any serious chances until the final 15 minutes.
Lacazette opened up his body once played through, but he could not add to his pre-season goals tally as he slotted wide.
Fourth-choice Chelsea keeper Marcin Bulka, a late sub in place of the barely-tested Caballero, then produced a wonderful save down low to somehow keep out Iwobi's close-range shot.
Despite Bulka's heroics, though, with the young keeper also blocking a powerful Lacazette drive, Arsenal's French striker converted from a few yards out at the death after Reiss Nelson squared the ball into his path.
The game was therefore decided on penalties, with Ruben Loftus-Cheek - back in action for the first time since his World Cup exploits with England - missing the only spot kick as Arsenal came out on top 6-5.
CHELSEA XI: Caballero (Bulka 78'); Azpilicueta (Zappacosta 63'), Luiz (Christensen 46'), Rudiger (Ampadu 63'), Alonso (Emerson 63'); Fabregas (Drinkwater 46'), Jorginho (Loftus-Cheek 78'), Barkley (Bakayoko 78'); Pedro (Moses 63'), Morata (Abraham 63'), Hudson-Odoi (Piazon 85')
ARSENAL XI: Cech; Bellerin (Chambers 79'), Sokratis, Mustafi, Kolasinac (Maitland-Niles 70'); Elneny, Smith Rowe (Iwobi 63'), Guendouzi; Mkhitaryan (Nelson 72'), Aubameyang (Lacazette 72'), Ozil